Fractal Structure and Non Extensive Statistics

2012

Tsallis q-extension of statistics and fractal generalization of dynamics are two faces of the same physical reality, as well as the Kernel modern complexity theory. The fractal generalization dynamics is based at the multiscale – multifractal characters of complex dynamics in the physical space-time and the complex system’s dynamical phase space. Tsallis q-triplet of non-extensive statistics can be used for the experiment test of q-statistic as well as of the fractal dynamics. In this study we present indicative experimental verifications of Tsallis theory in various complex systems such as solar plasmas, (planetic magnetospheres, cosmic stars and cosmic rays), atmospheric dynamics, seismogenesis and brain dynamics.

Annals of the New York Academy of Sciences, 1985

EPJ Web of Conferences, 2018

The role played by non-extensive thermodynamics in physical systems has been under intense debate for the last decades. Some possible mechanisms that could give rise to non-extensive statistics have been formulated along the last few years, in particular the existence of a fractal structure in thermodynamic functions for hadronic systems. We investigate the properties of such fractal thermodynamical systems, in particular the fractal scale invariance is discussed in terms of the Callan-Symanzik equation. Finally, we propose a diagrammatic method for calculations of relevant quantities.

Fractal and Fractional

We introduce fractal Tsallis entropy and show that it satisfies Shannon–Khinchin axioms. Analogously to Tsallis divergence (or Tsallis relative entropy, according to some authors), fractal Tsallis divergence is defined and some properties of it are studied. Within this framework, Lesche stability is verified and an example concerning the microcanonical ensemble is given. We generalize the LMC complexity measure (LMC is Lopez-Ruiz, Mancini and Calbert), apply it to a two-level system and define the statistical complexity by using the Euclidean and Wootters’ distance measures in order to analyze it for two-level systems.

Annals of Physics, 2004

We discuss basic statistical properties of systems with multifractal structure. This is possible by extending the notion of the usual Gibbs-Shannon entropy into more general framework-R enyiÕs information entropy. We address the renormalization issue for R enyiÕs entropy on (multi)fractal sets and consequently show how R enyiÕs parameter is connected with multifractal singularity spectrum. The maximal entropy approach then provides a passage between R enyiÕs information entropy and thermodynamics of multifractals. Important issues such as R enyiÕs entropy versus Tsallis-Havrda-Charvat entropy and PDF reconstruction theorem are also studied. Finally, some further speculations on a possible relevance of our approach to cosmology are discussed.

Contemporary Mathematics, 2013

Fractals define a new and interesting realm for a discussion of basic phenomena in quantum field theory and statistical mechanics. This interest results from specific properties of fractals, e.g., their dilatation symmetry and the corresponding absence of Fourier mode decomposition. Moreover, the existence of a set of distinct dimensions characterizing the physical properties (spatial or spectral) of fractals make them a useful testing ground for dimensionality dependent physical problems. This paper addresses specific problems including the behavior of the heat kernel and spectral zeta functions on fractals and their importance in the expression of spectral properties in quantum physics. Finally, we apply these results to specific problems such as thermodynamics of quantum radiation by a fractal blackbody.

Physica A: Statistical Mechanics and its Applications, 1998

We apply the Principle of Maximum Entropy to the study of a general class of deterministic fractal sets. The scaling laws peculiar to these objects are accounted for by means of a constraint concerning the average content of information in those patterns. This constraint allows for a new statistical characterization of fractal objects and fractal dimension. PACS: 47.53.+n, 05.90.+m * Present address:

Physica A: Statistical Mechanics and its Applications, 2000

In this paper, we explain the multifractal and thermodynamical properties of deterministic fractal spectra, and compare the most relevant behaviors with those obtained for some natural spectra with multifractal properties. In particular, we compare the heat capacity, C(T ), derived from Cantor-set-type spectra with the ones obtained from several atomic spectra. We ÿnd that, although many of the properties of deterministic fractals, previously reported, are also reproduced in natural fractals (oscillations of C(T ) in log scale), new features appear in the natural case: the mean value of C(T ) is not always related to the multifractal properties, and the oscillations are present in cases for which they disappear in deterministic fractals.

Entropy, 2017

In classical concepts, theoretical models are built assuming that the dynamics of the complex system's stuctural units occur on continuous and differentiable motion variables. In reality, the dynamics of the natural complex systems are much more complicated. These difficulties can be overcome in a complementary approach, using the fractal concept and the corresponding non-differentiable theoretical model, such as the scale relativity theory or the extended scale relativity theory. Thus, using the last theory, fractal entropy through non-differentiable Lie groups was established and, moreover, the pairs generating mechanisms through fractal entanglement states were explained. Our model has implications in the dynamics of biological structures, in the form of the "chameleon-like" behavior of cholesterol.

Physical Review E, 2006

The specific heat corresponding to systems with deterministic fractal energy spectra is known to present logarithmic-periodic oscillations as a function of the temperature T in the low T region around a mean value given by a characteristic dimension of the energy spectrum. In general, it is considered that the presence of disorder does not affect strongly these results, and that the fractal structure of the energy spectrum dominates. In this paper, we study the properties of the specific heat derived from random fractal energy spectra as a function of the degree of disorder present in the spectra. To study the influence of the disorder, we analyze the specific heat using three different properties: the specific heat mean value and the periods and amplitudes of the oscillations of the specific heat around its mean value. By studying the distributions and the mean values of these three properties, we obtain that the disorder does not influence very much the mean value of the specific heat. However, concerning the behavior of periods and amplitudes, we obtain a critical value of the disorder present in the energy spectra. Below this critical value, we find a low effect of the disorder and quasideterministic behavior indicating that the fractal structure is the dominant effect, but above the critical value, the disorder dominates and the behavior of the specific heat is practically chaotic.